Subgroups — C₂₃×#33822 #19181448

Order 5888.

496 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3168	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2229	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2228	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2220	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2219	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2218	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#2211	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2210	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#529	—	2	2	no	—
11	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
13	C₂×C₂₃×D₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂₃×#499	C₂²	1	2	yes	—
17	C₂₃×#495	C₂²	1	2	yes	—
18	C₂₃×#488	—	2	2	no	—
19	C₂₃×#467	C₂²	1	2	yes	—
20	C₂₃×#466	C₂²	1	2	yes	—
21	C₂₃×#465	—	2	2	no	—
22	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
23	C₄×SD₁₆×C₂₃	—	2	2	no	—
24	C₄×D₈×C₂₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#367	C₂²	1	1	yes	characteristic
27	C₂₃×#367	C₂²	1	1	yes	characteristic
28	C₂₃×#366	C₂²	1	2	yes	—
29	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₂₃×#358	C₂²	1	2	yes	—
32	C₂₃×#358	C₂²	1	2	yes	—
33	C₂₃×#357	C₂²	1	2	yes	—
34	C₂₃×#357	C₂²	1	2	yes	—
35	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₂₃	—	2	2	no	—
37	C₂×D₈×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×#126	—	4	2	no	—
41	C₂₃×#125	—	2	2	no	—
42	C₂₃×#124	—	2	2	no	—
43	C₂₃×#123	—	4	2	no	—
44	C₂₃×C₂²⋊Q₈	—	2	2	no	—
45	C₂₃×#121	—	2	2	no	—
46	C₄×Q₈×C₂₃	—	2	2	no	—
47	C₄×D₄×C₂₃	—	2	2	no	—
48	C₄×D₄×C₂₃	—	4	2	no	—
49	C₂₃×#117	—	4	2	no	—
50	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
51	C₂₃×Q₃₂	—	4	1	no	—
52	C₂₃×SD₃₂	—	4	2	no	—
53	C₂₃×D₁₆	—	4	1	no	—
54	C₂₃×M₃₂	—	4	1	no	—
55	C₂₃×M₃₂	—	4	1	no	—
56	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
58	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
59	C₂₃×#107	C₂³	1	2	yes	—
60	C₂₃×#107	C₂³	1	2	yes	—
61	C₂₃×#106	—	2	2	no	—
62	C₂₃×#103	—	2	2	no	—
63	C₂₃×#103	—	2	2	no	—
64	C₂₃×#102	—	2	2	no	—
65	C₂₃×#102	—	2	2	no	—
66	C₄×C₈×C₂₃	D₄	1	2	yes	—
67	C₄×C₈×C₂₃	C₂³	1	2	yes	—
68	C₂×Q₈×C₂₃	—	2	2	no	—
69	C₂×D₄×C₂₃	—	2	2	no	—
70	C₂×D₄×C₂₃	—	4	2	no	—
71	C₂²×C₄×C₂₃	—	4	2	no	—
72	C₂²×C₄×C₂₃	D₈	1	2	yes	—
73	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₂₃	—	2	2	no	—
75	Q₁₆×C₂₃	—	2	2	no	—
76	SD₁₆×C₂₃	—	4	4	no	—
77	D₈×C₂₃	—	2	2	no	—
78	D₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	1	no	—
80	C₂×C₈×C₂₃	—	2	1	no	—
81	C₂×C₈×C₂₃	—	4	1	no	—
82	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	—	4	2	no	—
88	C₄⋊C₄×C₂₃	—	4	2	no	—
89	C₄⋊C₄×C₂₃	—	4	2	no	—
90	C₄⋊C₄×C₂₃	—	4	2	no	—
91	C₂²⋊C₄×C₂₃	—	4	2	no	—
92	C₂²⋊C₄×C₂₃	—	4	2	no	—
93	C₂²⋊C₄×C₂₃	—	4	2	no	—
94	C₂²⋊C₄×C₂₃	—	4	2	no	—
95	C₂²⋊C₄×C₂₃	—	4	2	no	—
96	C₄²×C₂₃	—	4	2	no	—
97	C₄²×C₂₃	D₈	1	1	yes	characteristic
98	C₄²×C₂₃	D₈	1	1	yes	characteristic
99	C₄²×C₂₃	C₂×D₄	1	2	yes	—
100	C₁₆×C₂₃	—	2	1	no	—
101	C₁₆×C₂₃	—	2	2	no	—
102	C₁₆×C₂₃	—	2	1	no	—
103	#33822	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	—	4	2	no	—
105	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
106	Q₈×C₂₃	—	4	2	no	—
107	Q₈×C₂₃	—	4	2	no	—
108	D₄×C₂₃	—	4	2	no	—
109	D₄×C₂₃	—	4	2	no	—
110	D₄×C₂₃	—	8	4	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	4	2	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	8	4	no	—
118	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
119	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₂₃	—	2	1	no	—
123	C₈×C₂₃	—	2	2	no	—
124	C₈×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₈×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
126	#3168	C₂×C₂₃	1	2	yes	complemented
127	#2229	C₂×C₂₃	1	2	yes	complemented
128	#2228	C₂×C₂₃	1	2	yes	complemented
129	#2220	C₂×C₂₃	1	2	yes	complemented
130	#2219	C₂×C₂₃	1	1	yes	characteristic, complemented
131	#2218	C₂×C₂₃	1	1	yes	characteristic
132	#2211	C₂×C₂₃	1	2	yes	complemented
133	#2210	C₂×C₂₃	1	2	yes	complemented
134	#2150	C₂×C₂₃	1	1	yes	characteristic, complemented
135	C₂²×C₂₃	—	2	2	no	—
136	C₂²×C₂₃	—	2	1	no	—
137	C₂²×C₂₃	—	8	2	no	—
138	C₂²×C₂₃	—	8	2	no	—
139	C₂²×C₂₃	—	8	2	no	—
140	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	8	2	no	—
145	C₄×C₂₃	—	8	2	no	—
146	C₄×C₂₃	—	8	2	no	—
147	C₄×C₂₃	#537	1	1	yes	characteristic
148	C₄×C₂₃	#575	1	1	yes	characteristic
149	#529	—	2	2	no	—
150	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
152	C₂×D₁₆	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
155	#499	C₂²×C₂₃	1	2	yes	—
156	#495	C₂²×C₂₃	1	2	yes	—
157	#488	—	2	2	no	—
158	#467	C₂²×C₂₃	1	2	yes	—
159	#466	C₂²×C₂₃	1	2	yes	—
160	#465	—	2	2	no	—
161	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
162	C₄×SD₁₆	—	2	2	no	—
163	C₄×D₈	C₂²×C₂₃	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
165	#367	C₂²×C₂₃	1	1	yes	characteristic
166	#367	C₂²×C₂₃	1	1	yes	characteristic
167	#366	C₂²×C₂₃	1	2	yes	—
168	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
169	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
170	#358	C₂²×C₂₃	1	2	yes	—
171	#358	C₂²×C₂₃	1	2	yes	—
172	#357	C₂²×C₂₃	1	2	yes	—
173	#357	C₂²×C₂₃	1	2	yes	—
174	C₂×C₂₃	—	4	1	no	—
175	C₂×C₂₃	—	16	2	no	—
176	C₂×C₂₃	#3245	1	1	yes	characteristic
177	C₂×C₂₃	#3414	1	2	yes	—
178	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
179	C₂×SD₁₆	—	2	2	no	—
180	C₂×D₈	C₂³×C₂₃	1	2	yes	—
181	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
183	#126	—	4	2	no	—
184	#125	—	2	2	no	—
185	#124	—	2	2	no	—
186	#123	—	4	2	no	—
187	C₂²⋊Q₈	—	2	2	no	—
188	#121	—	2	2	no	—
189	C₄×Q₈	—	2	2	no	—
190	C₄×D₄	—	2	2	no	—
191	C₄×D₄	—	4	2	no	—
192	#117	—	4	2	no	—
193	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
194	Q₃₂	—	4	1	no	—
195	SD₃₂	—	4	2	no	—
196	D₁₆	—	4	1	no	—
197	M₃₂	—	4	1	no	—
198	M₃₂	—	4	1	no	—
199	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
200	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
201	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
202	#107	C₂³×C₂₃	1	2	yes	—
203	#107	C₂³×C₂₃	1	2	yes	—
204	#106	—	2	2	no	—
205	#103	—	2	2	no	—
206	#103	—	2	2	no	—
207	#102	—	2	2	no	—
208	#102	—	2	2	no	—
209	C₄×C₈	D₄×C₂₃	1	2	yes	—
210	C₄×C₈	C₂³×C₂₃	1	2	yes	—
211	C₂₃	#33822	1	1	yes	characteristic, direct product factor, Sylow
212	C₂×Q₈	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	D₈×C₂₃	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
218	Q₁₆	—	2	2	no	—
219	Q₁₆	—	2	2	no	—
220	SD₁₆	—	4	4	no	—
221	D₈	—	2	2	no	—
222	D₈	—	2	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
228	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	D₈×C₂₃	1	1	yes	characteristic
242	C₄²	D₈×C₂₃	1	1	yes	characteristic
243	C₄²	C₂×D₄×C₂₃	1	2	yes	—
244	C₁₆	—	2	1	no	—
245	C₁₆	—	2	2	no	—
246	C₁₆	—	2	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	8	4	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
262	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
263	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
264	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	1	no	—
267	C₈	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
268	C₈	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	2	no	—
281	C₄	C₂₃×#537	1	1	yes	characteristic
282	C₄	C₂₃×#575	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	16	2	no	—
285	C₂	C₂₃×#3245	1	1	yes	characteristic
286	C₂	C₂₃×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂₃×#2218	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂₃×#3168	C₂	semidirect
2. C₂₃×#2229	C₂	semidirect
3. C₂₃×#2228	C₂	semidirect
4. C₂₃×#2220	C₂	semidirect
5. C₂₃×#2219	C₂	semidirect
7. C₂₃×#2211	C₂	semidirect
8. C₂₃×#2210	C₂	semidirect
9. C₂₃×#2150	C₂	semidirect
103. #33822	C₂₃	direct (N × H)
126. #3168	C₂×C₂₃	semidirect
127. #2229	C₂×C₂₃	semidirect
128. #2228	C₂×C₂₃	semidirect
129. #2220	C₂×C₂₃	semidirect
130. #2219	C₂×C₂₃	semidirect
132. #2211	C₂×C₂₃	semidirect
133. #2210	C₂×C₂₃	semidirect
134. #2150	C₂×C₂₃	semidirect
211. C₂₃	#33822	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3168	2
C₂₃×#2229	2
C₂₃×#2228	2
C₂₃×#2220	2
C₂₃×#2219	2
C₂₃×#2218	2
C₂₃×#2211	2
C₂₃×#2210	2
C₂₃×#2150	2
#33822	23

Subgroups — C23×#33822 #19181448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33822 #19181448